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I was thinking about orientations (of a vector space, of a simplex, ...) and how there seem to always be exactly two orientations. Moreover, orientations are defined "up to the parity of the permutation". So, eg., an orientation of a vector space seems to be not a basis, but an equivalence class of bases (where two bases are equivalent iff they differ by a permutation of the same parity; now, since there's two possible parities, there's two possible equivalence classes, hence two possible orientations).

What is the "formal" way to state this fact in terms of groups, quotient groups, and (perhaps) group actions? (Is it even a fact, to begin with?) I suspect that the answer involves the terms: symmetric group, alternating group, quotient group, $\Bbb Z / 2\Bbb Z$, and maybe group action and orbit.

What group is responsible for the definition/construction of orientation? I think it has to be a quotient group of the symmetric group $S_n$ (say, for a vector space of dimension $n$), because you start with a basis and take all possible permutations and then consider the permutations that differ by (say) an even permutation to have the same orientation. (But what's the formal way to state this?)

Group actions seem to be able to induce quotient spaces; eg. Wikipedia says that an orbifold looks (locally) like the the quotient space of Euclidean space under the (linear) action of a finite group. It also says that a manifold with boundary is an orbifold because it's the quotient of its double by an action of $\Bbb Z / 2\Bbb Z$.

And I think that the group special linear group $\text{SL}(2,\Bbb Z)$ and some of its subgroups act on the the upper half (complex) plane $\Bbb H$ and so can induce quotient topological spaces.

What is the proper way to phrase/understand how groups and/or group actions induce quotients on sets / topological spaces / manifolds / stuff? Is it the group action that induces the quotient, or the group itself?

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    $\begingroup$ I believe if the group action is faithful there's not much difference between being induced by the action or the group itself. If the action is not faithful quotients can be induced by the kernel of the action too. When an action isn't faithful in well studied contexts usually people take a quotient of the group to make the action faithful. I believe this is why we consider modular forms to be in $SL(2,\Bbb Z)$ instead of $GL(2,\Bbb Z)$. So the difference may be hidden in well studied contexts. $\endgroup$ – N8tron Jun 7 '18 at 12:48
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Each orientation is an orbit space of the action of the alternating group $A_n$ (the group of even permutations) on your set of vectors or vertices or whatever you're orienting. There are two orbits because $A_n$ has index 2 as a subgroup of the group of permutations $S_n$.

In other cases, for example, oriented manifolds, orientations are defined as forms that never vanish, and two of those forms are equivalent if one is a positive multiple (by a smooth function) of the other. Since there are only two signs and they can never vanish, it gives exactly two orientations when the manifold is connected.

In general, you can consider the group homomorphism $S_n\to\{-1,1\}\cong \mathbb{Z}/2\mathbb{Z}$ asigning to each permutation its sign, whose kernel is precisely $A_n$. Hence, there is a relation between $A_n$ and $\mathbb{Z}/2\mathbb{Z}$.

For general understanding of group actions this question might help you. Once you understand how group actions determine quotients of sets, then the additional structures are usually canonical.

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